A369695 -id:A369695 - cp's OEIS frontend

A369696 Number of unordered pairs (p,q) of partitions of n such that the set of parts in q is equal to the set of parts in p.

1, 1, 2, 3, 5, 8, 12, 19, 27, 42, 61, 91, 130, 192, 271, 401, 556, 802, 1126, 1597, 2217, 3132, 4315, 6003, 8257, 11370, 15527, 21251, 28798, 39043, 52722, 70911, 95047, 127155, 169431, 225072, 298362, 393946, 519294, 682090, 894251, 1168258, 1524370, 1981554

Offset: 0

Alois P. Heinz, Jan 29 2024

nonn

			a(5) = 8: (11111, 11111), (2111, 2111), (2111, 221), (221, 221), (311, 311), (32, 32), (41, 41), (5, 5).

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A000041, A297388, A369695, A369697.

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
      b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
    end:
a:= n-> (b(n$3)+combinat[numbpart](n))/2:
seq(a(n), n=0..50);

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1] + Sum[Sum[b[Sequence @@ Sort[{n-i*j, m-i*h}], i-1], {h, 1, m/i}], {j, 1, n/i}]]];
a[n_] := (b[n, n, n] + PartitionsP[n])/2;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

a(n) = (A000041(n) + A369695(n))/2.

A369697 Number of unordered pairs (p,q) of distinct partitions of n such that the set of parts in q is equal to the set of parts in p.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 5, 12, 19, 35, 53, 91, 136, 225, 325, 505, 741, 1107, 1590, 2340, 3313, 4748, 6682, 9412, 13091, 18241, 25080, 34478, 47118, 64069, 86698, 117012, 157121, 210189, 280385, 372309, 493279, 650905, 856913, 1123675, 1471196, 1918293, 2497470

Offset: 0

Alois P. Heinz, Jan 29 2024

nonn

			a(5) = 1: (221, 2111).
a(6) = 1: (2211, 21111).
a(7) = 4: (22111, 211111), (2221, 211111), (2221, 22111), (331, 31111).
a(8) = 5: (221111, 2111111), (22211, 2111111), (22211, 221111), (3221, 32111), (3311, 311111).
a(9) = 12: (2211111, 21111111), (222111, 21111111), (222111, 2211111), (22221, 21111111), (22221, 2211111), (22221, 222111), (32211, 321111), (33111, 3111111), (3321, 321111), (3321, 32211), (4221, 42111), (441, 411111).
a(10) = 19: (22111111, 211111111), (2221111, 211111111), (2221111, 22111111), (222211, 211111111), (222211, 22111111), (222211, 2221111), (322111, 3211111), (32221, 3211111), (32221, 322111), (331111, 31111111), (33211, 3211111), (33211, 322111), (33211, 32221), (3331, 31111111), (3331, 331111), (42211, 421111), (4411, 4111111), (442, 4222), (5221, 52111).

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A000041, A297388, A369695, A369696.

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
      b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
    end:
a:= n-> (b(n$3)-combinat[numbpart](n))/2:
seq(a(n), n=0..50);

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1] + Sum[Sum[b[Sequence @@ Sort[{n-i*j, m-i*h}], i-1], {h, 1, m/i}], { j, 1, n/i}]]];
a[n_] := (b[n, n, n] - PartitionsP[n])/2;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

a(n) = (A369695(n) - A000041(n))/2.

cp's OEIS Frontend

A369696 Number of unordered pairs (p,q) of partitions of n such that the set of parts in q is equal to the set of parts in p.

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